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Properties

Label 2-3e6-9.7-c1-0-10
Degree $2$
Conductor $729$
Sign $0.5 - 0.866i$
Analytic cond. $5.82109$
Root an. cond. $2.41269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.400 − 0.694i)2-s + (0.678 + 1.17i)4-s + (1.37 + 2.38i)5-s + (−1.18 + 2.05i)7-s + 2.69·8-s + 2.20·10-s + (0.125 − 0.216i)11-s + (−1.30 − 2.26i)13-s + (0.952 + 1.64i)14-s + (−0.279 + 0.483i)16-s − 0.293·17-s − 2.78·19-s + (−1.86 + 3.23i)20-s + (−0.100 − 0.173i)22-s + (3.34 + 5.79i)23-s + ⋯
L(s)  = 1  + (0.283 − 0.490i)2-s + (0.339 + 0.587i)4-s + (0.614 + 1.06i)5-s + (−0.449 + 0.778i)7-s + 0.951·8-s + 0.696·10-s + (0.0377 − 0.0653i)11-s + (−0.362 − 0.627i)13-s + (0.254 + 0.440i)14-s + (−0.0697 + 0.120i)16-s − 0.0711·17-s − 0.638·19-s + (−0.417 + 0.722i)20-s + (−0.0213 − 0.0370i)22-s + (0.697 + 1.20i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(729\)    =    \(3^{6}\)
Sign: $0.5 - 0.866i$
Analytic conductor: \(5.82109\)
Root analytic conductor: \(2.41269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{729} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 729,\ (\ :1/2),\ 0.5 - 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74915 + 1.00987i\)
\(L(\frac12)\) \(\approx\) \(1.74915 + 1.00987i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (-0.400 + 0.694i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.37 - 2.38i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.18 - 2.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.125 + 0.216i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.30 + 2.26i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.293T + 17T^{2} \)
19 \( 1 + 2.78T + 19T^{2} \)
23 \( 1 + (-3.34 - 5.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.177 + 0.307i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.38 + 2.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6.99T + 37T^{2} \)
41 \( 1 + (-4.85 - 8.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.130 - 0.225i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.71 + 9.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 5.43T + 53T^{2} \)
59 \( 1 + (2.98 + 5.17i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.92 + 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.905 + 1.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.370T + 71T^{2} \)
73 \( 1 - 5.02T + 73T^{2} \)
79 \( 1 + (0.401 - 0.695i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.37 + 2.38i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.4T + 89T^{2} \)
97 \( 1 + (-7.41 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.63933010100292164624011378816, −9.899099248478150795864123135592, −8.933224987767846173532237561794, −7.83357220131329628101313114990, −6.97316984325763654507973431528, −6.18297440927681242747789210024, −5.15547847906008289115977976213, −3.66044930295210540635180957981, −2.85492097412194394475470933253, −2.07403319719057158188113461016, 0.976260125419474848159466710257, 2.25313220721820892132585390267, 4.12150402274051918788411288245, 4.89681523958897741147177987465, 5.77040168168177846968902126245, 6.72175054790640946524482015491, 7.32127519905739670472690082463, 8.686760373187718202627002250832, 9.324474860453348254281556340565, 10.34682223046345862167063390575

Graph of the $Z$-function along the critical line